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Tingyu Qu, Michele Masseroni, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Barbaros Özyilmaz, Thomas Ihn, Klaus Ensslin

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[Observation of weak localization in dual-gated bilayer <math>  <mrow>    <mi>Mo</mi>    <msub>      <mi>S</mi>      <mn>2</mn>    </msub>  </mrow></math>](https://mdr.nims.go.jp/datasets/9317ac97-f5be-44e7-8007-acff0cbc1d9e)

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Observation of weak localization in dual-gated bilayer ${\rm{Mo}}{{{\rm{S}}}_2}$PHYSICAL REVIEW RESEARCH 6, 013216 (2024)Observation of weak localization in dual-gated bilayer MoS2Tingyu Qu ,1,2,* Michele Masseroni ,3 Takashi Taniguchi ,3 Kenji Watanabe ,3 Barbaros Özyilmaz,1,2Thomas Ihn ,4 and Klaus Ensslin 41NUS Graduate School, Integrative Sciences and Engineering Programme, National University of Singapore, Singapore2Department of Physics, National University of Singapore, Singapore3National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan4Solid State Physics Laboratory, ETH Zürich, 8093 Zürich, Switzerland(Received 23 September 2023; accepted 31 January 2024; published 28 February 2024)We investigate the magnetoresistance of a dual-gated bilayer MoS2 encapsulated by hexagonal boron nitride.At low magnetic fields (|B| < 0.5 T), we observe a negative magnetoresistance, which we identify as the weaklocalization effect. We determine both the phase coherence length and mean free path as a function of electrondensity and displacement field. Both characteristic lengths show a similar monotonic increase with electrondensity, while they are not affected by the displacement field. We further investigate the dephasing mechanismby measuring the temperature dependence of the phase coherence length. Our results suggest that when onlythe lower spin-orbit split bands (K ↑, K ′ ↓) contribute to transport is Coulomb scattering the dominant sourceof decoherence, while intervalley scattering seems not to play a relevant role in this regime. This observation isconsistent with the picture of spin-polarized valleys (spin-valley locking), where the intrinsic spin-orbit couplingprotects the spin states, rather than introducing an additional dephasing mechanism as in other materials.DOI: 10.1103/PhysRevResearch.6.013216I. INTRODUCTIONIn the diffusive transport regime, the magnetoresistance(MR) of electronic systems deviates from Boltzmann theoryand displays an enhancement at zero field [1]. This mag-netoresistance peak is a quantum effect that originates fromtime-reversed paths of electrons interfering constructively inscattering loops. This effect is referred to as weak localization(WL) [2].Transition metal dichalcogenide (TMDC) monolayers haveno inversion symmetry in the lattice sites [3,4]. Furthermore,spin-orbit coupling is particularly strong in the valence band,but also present in the conduction band. For example, molyb-denum disulfide (MoS2) hosts an intrinsic spin-orbit coupling(SOC) that behaves as an out-of-plane Zeeman field and pinsthe two spins with opposite directions in K and K′ valleys[5,6]. In contrast to other materials (e.g., Bernal stackedbilayer graphene, bilayer or trilayer MoS2), the K-valley elec-trons of different layers behave as independent systems owingto the weak interlayer coupling [7,8].In the presence of SOC effects, the spin of an electronrotates as it is scattered between self-crossing paths, yieldinga destructive interference and a lower zero-field resistivitycalled weak antilocalization (WAL) [9,10]. The study of*ty.qu@nus.edu.sgPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.W(A)L in two-dimensional (2D) systems in a magnetic fieldperpendicular to the 2D plane allows us to assess some fun-damental characteristics of the charge carriers, such as phasecoherence and scattering rates as well as the strength of SOC[11–14].Previous investigations about the W(A)L in monolayer[15], bilayer [16], and few-layer MoS2 samples [17,18] lacka dual-gated device structure. In addition, the influence ofcarrier population of different spin-split bands in the presenceof W(A)L has not been presented yet. Also, in light of the lim-itation that the Hikami-Larkin-Nagaoka (HLN) model [9] isinsufficient to allow interpretation of the spin splitting due tothe intrinsic SOC in TMDC [12], caution might be warrantedwhen extracting spin relaxation lengths from the spin-orbitscattering in the HLN model.In this paper, we show that electron transport in a dual-gated bilayer MoS2 can be entirely described by WL, withouttaking into account spin-orbit coupling. We observe a pro-nounced WL peak when the electrons populate a single layer(top layer), while it is not present when both layers are occu-pied. We do not observe WAL, suggesting a minor effect ofSOC on quantum interference. We simplify the fitting modelby neglecting the SOC term in the HLN model and describeour data with only two parameters, namely, the phase coher-ence length and elastic mean free path. We determine themean free path from the zero-field conductivity and extractthe phase coherence length by fitting the weak localizationpeak, taking the phase coherence length as the only fittingparameter. Based on our evaluation, the phase coherencelength at a temperature of 1.3 K exceeds 100 nm for densities>7×1012 cm−2 and it shows a linear dependence on thedensity. On the other hand, when tuning the displacements2643-1564/2024/6(1)/013216(8) 013216-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-4409-3072https://orcid.org/0000-0003-1663-8239https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-5587-6953https://orcid.org/0000-0001-7007-6949https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.6.013216&domain=pdf&date_stamp=2024-02-28https://doi.org/10.1103/PhysRevResearch.6.013216https://creativecommons.org/licenses/by/4.0/TINGYU QU et al. PHYSICAL REVIEW RESEARCH 6, 013216 (2024)FIG. 1. (a) Device schematics. The bilayer MoS2 is encapsulated between two h-BN flakes, with metallic contacts (Au) as the source/drain(S/D) and dual gates (Au as top gate and graphite as bottom gate). (b) Phase diagram for the population of the carriers in different bands asa function of top and bottom gates. The diagram is reconstructed from the measurement of the magnetoresistance at T = 1.3 K and B = 7 Tpresented in Ref. [19]. The white-color regime refers to the insulating state. The colored regimes (from left to right, labeled by 1–4) indicatethe population of the lower band in the top layer, upper band in the top layer, lower band in the bottom layer, and upper band in the bottomlayer, respectively. (c) σxx − ns (top subfigure) and Le-ns (bottom subfigure) curves at T = 1.3 K and B = 0 T, based on the forms σ = nseμand Le = σ h̄/e2√4π/(gvns ) (where gv = 2 is the valley degeneracy in each spin-valley locked band). A kink in σxx-ns can be identified by theintersection of the two linear fits. The onset of mobility in the upper spin-split band correspond to 3.8×1012 cm−2. (d) Normalized MR as afunction of B and ns populated in the first layer at T = 1.3 K. The overlaid linecut shows the normalized MR at ns = 3.3×1012 cm−2.field while keeping the density constant, we do not observeany evident dependence. The temperature dependence of thephase coherence length shows T −α with α ≈ 0.5 in the single-band regime. All these observations hint towards Coulombinteraction as the main dephasing mechanism and a spin re-laxation length much longer than the phase coherence lengthin 2D MoS2.II. RESULTS AND DISCUSSIONOur device schematics is shown in Fig. 1(a). The bilayerMoS2 sample is encapsulated by two hexagonal boron nitrideflakes (h-BN) as the dielectric layers, which allows us totune the total carrier density (ns) from a top gate (Au) and abottom gate (graphite) by the parallel-capacitance model withthe form of ns = 1e (CBVBG + CT VTG), where CB = ε0ε/dB andCT = ε0ε/dT refer to the capacitances of the bottom BN andtop BN layers, respectively (ε0 and ε are the vacuum permit-tivity and the dielectric constant ∼3.3, respectively; dB and dTrefer to the thicknesses of the bottom and top BN layers); VBGand VTG refer to the back-gate and top-gate voltages, respec-tively. The details of the device fabrication and measurementare shown in Appendix A. Each MoS2 layer contributes withtwo spin-split bands, where the spin and valley are lockedbecause of the intrinsic spin-orbit coupling [5]. As reportedin our previous work [19], the onset of the population in eachspin-split band is determined from Shubnikov–de Haas (SdH)oscillation frequencies. To highlight the population of thecarriers in each band as a function of both gates, a schematicphase diagram is presented in Fig. 1(b). The four coloredregimes (from left to right) indicate the population of thecarriers from the top layer (labeled by 1 and 2) to the bottomlayer (labeled by 3 and 4) as the dual gating effect increases.The carrier density dependence of the conductivity(σ ) and mean free path (Le) at zero field is shown inFig. 1(c). The longitudinal conductivity σxx is given by σxx =ρxx/(ρxx2 + ρxy2), where ρxx and ρxy are the measured lon-gitudinal and Hall resistivities, respectively (see more detailsin Appendix B). We extract σxx ≈ 0.17 mS (over four timesof e2/h) at the lowest carrier density we could reach for theWL analysis (ns = 2.3×1012 cm−2). An approximate linearcorrelation between σxx and ns exists before and after a kinkpoint (at about ns = 3.8×1012 cm−2). The kink refers to theonset of mobility (μ) in the upper band, which arises from theonset of the population of carriers. The Le-ns curves at variousdisplacement fields (Dd ) show nearly identical patterns thatcan be described by a power law. Therefore, we conclude thatthe mean free path does not depend on the displacement field.Figure 1(d) shows a map of the normalized magnetoresistance[�MR = MR(B) − MR(B = 0 T)] as a function of B and nsin the regime where only one layer is populated. At basetemperature, clear zero-field peaks in the MR are observedin the entire single-layer regime (from ns = 2.3×1012 cm−2to ns = 8.0×1012 cm−2), demonstrating the relevance of the013216-2OBSERVATION OF WEAK LOCALIZATION IN … PHYSICAL REVIEW RESEARCH 6, 013216 (2024)quantum interference effect. The origin of the positive MR atlarge field in the single-band regime is discussed separately inour previous work [19].We turn our attention to analysis of W(A)L. First, wepresent the HLN model that includes the phase coherence,spin orbit, and elastic scattering in the form [1,9,10]�σ (B) = e22πh[ln(BφB)−  (12+ BφB)]+ e2πh[ln(BSO + BeB)−  (12+ BSO + BeB)]+ 3e22πh[− ln(4/3BSO + BφB)+  (12+ 4/3BSO + BφB)],where Bφ , BSO, Be refer to the field scales for the phase coher-ence, spin-orbit coupling, and elastic scattering, respectively.�σ (B) = σ (B) − σ (B = 0 T) represents the normalized con-ductivity in the unit of e2/πh.   is the digamma function.To understand the role of SOC on WAL, we simulatethe turnover from WL to WAL by varying BSO based onthe above HLN model. To satisfy the single-band structurerequired in the HLN model [12], we choose the density ns =3.3×1012 cm−2, for which we experimentally establish thatthe carriers occupy only the lower spin-orbit split bands inthe top layer. In the simulation, a turnover from WL to WALappears only if the length scale satisfies LSO < Lφ , whereLSO and Lφ are the spin relaxation length and phase coher-ence length, respectively (see Appendix B). However, in theexperiment, we do not observe WAL throughout the entiretunable range of the carrier density (from 2.3×1012 cm−2to 2.0×1013 cm−2). The crossover from WL to WAL wasobserved at high carrier densities (∼1.0×1014 cm−2) with anionic liquid gate [18], and it was shown that the spin-orbitlength diverges at densities lower than 5.0×1013 cm−2 [18].In our case, the absence of WAL implies LSO > Lφ and LSO �Le. Consequently, the values of LSO deduced from WL areless reliable than those deduced from WAL [18]. Moreover,our observation suggests that the intrinsic SOC in MoS2 hasa minor consequence for quantum interference of conductionelectrons, in agreement with the reported surprisingly long-lived and coherent spin dynamics in monolayer MoS2 [20,21],and so it has little impact on the shape of the magnetoconduc-tivity if LSO � Le, Lφ [2].Therefore, we simplify the HLN model by leaving out theSOC term, which allows us to accurately evaluate Lφ . By sub-stituting Bφ = h̄/(4eLφ2) and Be = h̄/(4eLe2), we describethe WL only with Lφ and Le in the form�σ (B) = Ne2πh[ (12+ h̄4eBLφ2)−  (12+ h̄4eBLe2)+ 2ln(LφLe)],where   is the digamma function and N is a coefficient thatimplies the number of populated bands in the system. If theelectrons are populated in the single band and the spin-orbitscattering is weak, N is equal to 1.Next, it should be noted that the accuracy of fitting theWL depends on the magnetic field scale. Specifically, L∅ isdetermined by the curvature of the MR near zero field, whileLe is determined by the larger field scale Be = h̄/(2eLe2),where the WL disappears completely [2]. We observe a pos-itive MR background at large magnetic field (consistent withthe observations in former studies [15,16]) that we attributeto the strong localization effect [19]. The presence of thisbackground limits the magnetic field range that we considerfor the fits. As a result, Be cannot be accurately determined, sowe determine Le from the zero-field conductivity and use thephase coherence term as the only fitting parameter to extractLφ . We select the MR section with the strongest curvature nearzero field (where both SOC and elastic scattering play a minorrole) with a scale Bφ = h̄/(4eLφ2) [see Fig. 2(a)] and fixLe = σ h̄/e2√4π/(gvns) with each ns for the fitting. Severalexamples of our fitting curves are shown in Fig. 2(b). As ns in-creases from 3.3×1012 cm−2 (lower band) to 4.3×1012 cm−2(upper band), there is a prominent increase of the slope atthe field range within Bφ , revealing a general trend that Lφincreases with ns.We then plot the calculated Le and extracted Lφ fromthe HLN model as a function of ns. As shown in Fig. 2(c),both Lφ and Le present a positive correlation with ns. Inthe double-band regime, we compare the estimated values ofLφ with N equal to 1 and 2 (for degenerate double bands),where qualitatively good fits are obtained in both cases forthe carrier density dependence, despite marginal differencesin the extracted Lφ . Also, the two subbands are not degeneratedue to the SOC. Once the upper spin-split band is populated,additional scattering sources such as intravalley scattering areinvolved. In such case, N should be less than 2 but the exactvalue is unknown. Therefore, in the following text, we keepthe value of N equal to 1 for the consistency of our fitting. Cer-tain limitations of our model for the double-band regime willbe drawn at the end of the section. Lφ increases from ∼60 nmat ns = 3.3×1012 cm−2 to ∼105 nm at ns = 7.4×1012 cm−2,which is among the largest reported for MoS2 [15–18]. Thisis attributed to a higher sample quality, as confirmed by thelarge electron mobility μ = 2300 cm2/Vs [19]. The ratio ofthe coherence length over mean free path (Lφ/Le) decreasesfrom above 4.0 at ns < 3.6×1012 cm−2 to below 2.0 at ns >6.3×1012 cm−2, where the WL peak disappears. Indeed, theWL effect is expected to disappear if the coherence length ison the order of the scattering length. In Fig. 2(d), we test therole of the displacement field on the phase coherence length.The displacement field can be used to tune the SOC strengthin material with an in-plane SOC such as Rashba [22,23].However, in our case, the displacement field plays a minor rolein both Lφ and Le at fixed ns in both single-band and double-band regimes. The weak tunability by the displacement fieldsuggests that extrinsic (Rashba-type) SOC, if present, playsonly a marginal role.We finally investigate the dephasing mechanism by fittingthe Lφ as a function of temperature. Figure 3(a) shows the tem-perature dependence of �σ (B) at ns = 3.0×1012 cm−2. Astemperature increases from 1.3 to 5.0 K, an obvious decreaseof the slope of �σ (B) within the interval |B| < Bφ is observeddue to the reduced Lφ , whereas Le remains roughly con-stant in the temperature range considered in our experiment013216-3TINGYU QU et al. PHYSICAL REVIEW RESEARCH 6, 013216 (2024)FIG. 2. (a) Symmetrized magnetoconductivity �σ (B), where �σ (B) = [σ (B) + σ (−B)]/2−σ (B = 0 T), in unit of e2/πh at ns =3.3×1012 cm−2 and T = 1.3 K. The phase coherence field (Bφ) is scaled in the region with the strongest curvature. (b) Examples aboutthe fitting of �σ (B) curves for both the single-band and double-band regimes. The black lines at the range of Bφ are the fits using our redefinedHLN model. (c). Top subfigure: Lφ and Le as a function of ns. The dashed line represents the fitting result with N equal to 2 in the double bandregime as the lower limit for the extracted Lφ . Bottom subfigure: Lφ/Le as a function of ns. Data measured at fixed Dd = 1 V/nm and T = 1.3K. (d) Lφ and Le as a function of Dd at ns = 2.5×1012 cm−2 (single band) and ns = 5.4×1012 cm−2 (double bands), measured at T = 1.3 K.The error bars in (c) and (d) are measured by a confidence interval of 99.7%.FIG. 3. (a) �σ (B) curves at different temperatures (from 1.3 to 5.0 K) in the lower band regime. (b) Le-T curves at different ns from2.3×1012 cm−2 (lower band) to 6.6×1012 cm−2 (upper band). (c) Lφ-T curves at different ns. The red solid lines are the fits using Lφ ∝ T −α .(d) The extracted α as a function of ns.013216-4OBSERVATION OF WEAK LOCALIZATION IN … PHYSICAL REVIEW RESEARCH 6, 013216 (2024)[see Fig. 3(b)]. A weak temperature dependence of Le isexpected when the resistance is dominated by impurity scat-tering. The slight decreasing trend with increased temperaturefor all ns is attributed to the metallic phase of MoS2. In mostcases, the Le-T curve appears to be nonmonotonic at T < 2 K.In fact, such an effect is more prominent the lower the den-sity (ns � 3.6×1012 cm−2), which hints at other spuriouseffects such as strong localization. Figure 3(c) depicts theLφ-T curves at different ns, which can fit into a power law inthe form of Lφ ∝ T −α , where α is a constant that depends onthe dephasing mechanism. We find that α is close to 0.5 whenonly the lower spin-orbit split band (K ↑, K ′ ↓) is occupied,whereas it decreases to around 0.35 when also the upper band(K ↑, K ′ ↓) is filled with electrons [as shown in Fig. 3(d)]. Inthe low carrier density regime, the α estimated by our fittingapproach hints towards e-e interaction as the main sourceof decoherence [24]. In the higher carrier density regime,our model is mainly limited by three factors. First, the exactnumber of channels (or value of N) in our fitting model isunknown because of the two nondegenerate spin-split bands.Second, additional dephasing sources such as intervalley scat-tering might play a role on the estimated value of Lφ . Third,the change of the scale in Lφ from 1.5 to 5.0 K is not largeenough to extract the precise value of α. Though the exactscattering mechanism cannot be extracted, our fitting showsa reasonable estimation of Lφ (up to 100 nm) and reveals aspin relaxation time much longer than the coherence time.Our result suggests that the spins are well preserved despiteseveral possible scattering events (e.g., Coulomb interaction,intravalley scattering) in atomically thin MoS2III. CONCLUSIONIn summary, we study weak localization with reliable con-trol of the carrier population in different spin-split bands ina dual-gated bilayer MoS2. We did not observe weak an-tilocalization in the attainable carrier density range. This isattributed to the pronounced intrinsic spin-orbit coupling inMoS2, which gives rise to a spin relaxation length muchlonger than the mean free path and the coherence length,in agreement with the reported long-lived spin relaxation inMoS2. Therefore, awareness should be warranted that HLNmodel cannot reliably estimate the spin relaxation length onlyin the presence of weak localization. Nevertheless, our modelcan be expanded close to zero field to estimate the phasecoherence length. From our analysis, we could extract phasecoherence lengths of the order 100 nm, in agreement withprevious publications. At low densities, when only the lowerspin-orbit split bands are occupied, the coherence length islimited by electron-electron scattering.ACKNOWLEDGMENTSWe thank S. Iwakiri and Z. Lei for fruitful discussions.We thank P. Märki, T. Bähler, as well as the FIRST staff fortheir technical support. T.I. and K.E. acknowledge supportfrom the European Graphene Flagship Core3 Project, SwissNational Science Foundation via NCCR Quantum Science,and H2020 European Research Council (ERC) Synergy Grantunder Grant Agreement No. 95154. B.O. acknowledges thesupport from the Singapore NRF Investigatorship (Grant No.NRF-NRFI2018-8), Competitive Research Programme (GrantNo. NRF-CRP22-2019-8), and MOE-AcRF-Tier 2 (GrantNo. MOE-T2EP50220-0017). K.W. and T.T. acknowledgesupport from JSPS KAKENHI (Grants No. 19H05790, No.20H00354, and No. 21H05233).APPENDIX A: DEVICE FABRICATIONAND MEASUREMENTSFirst, we assemble the bottom hBN (as the bottom di-electric layer) and graphite flake (as the bottom gate) ontoa Si/SiO2 substrate (with 285-nm-thick oxide layer), usinga standard dry-transfer method [7]. We then pattern metalliccontacts (made by Cr/Au: 5 nm/10 nm) through standard elec-tron beam lithography and electron beam evaporation. Theresiduals on the contact regions (due to lithography process)are then thoroughly cleaned by the tip of an atomic forcemicroscope in contact mode. Next, we identify the bilayerMoS2. We employ the mechanical exfoliation from bulk MoS2crystals (obtained from SPI Supplies) onto a Si/SiO2 sub-strate (with 285-nm-thick oxide layer) and then identify thebilayer by its optical contrast, with a highly reliable method[25,26]. Then, we assemble the second stack containing thetop hBN (as the top dielectric layer) and the selected bi-layer MoS2. This stack is aligned and transferred onto thecontacts. The transfer is performed in a glove box with anargon atmosphere, ensuring minimal exposure to H2O andO2 (both <0.1 ppm). Finally, a metallic top gate (that coversthe entire MoS2 flake) is patterned using standard electronbeam lithography and electron beam evaporation. The finalstack is annealed in vacuum condition at 250 °C for 4 h toreduce the bubbles and improve the quality of the contactinterface. The thicknesses for the bottom and top hBN layersare determined by atomic force microscopy in tapping mode,which yields dB = 13 nm and dT = 20 nm, respectively. Thedevice information is shown in Fig. 4.According to a parallel plate capacitor model (C = εε0/d),the thickness dB (dT ) yields the capacitance per area CB =225 nF/cm2 (CT = 146 nF/cm2). These values are used inthe main text to determine the electron density as a functionof bottom and top gate voltages. The capacitance per areaobtained by the parallel plate capacitor model is then con-firmed experimentally by measuring the electron density fromthe SdH oscillations [see Fig. 2(b) of Ref. [19] for a directcomparison].The measurements are performed by lock-in techniques(with excitation voltage at 100 µV and frequency at ∼30 Hz).The temperature range is 1.3–10 K and magnetic-field rangeis 0–7 T.APPENDIX B: SIMULATION ON THE TURNOVERFROM WL TO WALThe general form for weak location in thin films can bewritten by [1]σ (H ) = − e22π2h[ (12+ B1B)−  (12+ B2B)+ 12 (12+ B3B)− 12 (12+ B4B)],013216-5TINGYU QU et al. PHYSICAL REVIEW RESEARCH 6, 013216 (2024)FIG. 4. (a) Optical micrograph of the stack without the top gate (TG). A multilayer graphite (outlined by the red line) acts as the back gate(BG); the bottom BN (bBN, outlined by the red line) acts as bottom dielectric layer; the bilayer MoS2 (outlined by the blue line) is probedby Au contacts (one outlined by the black line); the top BN (tBN, outlined by the purple line) acts as the top dielectric layer. (b) Opticalmicrograph of the stack after forming the top gate. (c) Thickness profile (∼13.5 nm) of the bottom hBN flake (in the region denoted by thepurple box) measured by atomic force microscopy. (d) Thickness profile (∼20 nm) of the top hBN flake (in the region denoted by the greenbox) measured by atomic force microscopy.whereB1 = Bo + Bso + Bs,B2 = 43 Bso + 23 Bs + Bi,B3 = 2Bs + Bi,B4 = 43 BSO + 23 Bs + Bi.The terms o, i, s, and SO in the above formulae refer topotential scattering (for mean free path), inelastic scattering(for phase decoherence), magnetic scattering, and spin-orbitscattering (for spin relaxation related to spin-orbit coupling),respectively.   is the digamma function.In pristine MoS2, the magnetic scattering should be leftout, leading to the Hikami-Larkin-Nagaoka (HLN) modelthat includes the spin-orbit term, phase-coherence term, andelastic-scattering term in the form�σ (B) = e22πh[ln(BφB)−  (12+ BφB)]+ e2πh[ln(BSO + BeB)−  (12+ BSO + BeB)]+ 3e22πh[− ln(4/3BSO + BφB)+  (12+ 4/3BSO + BφB)],where BSO, Bφ , Be refer to the field scales for the spin-orbitcoupling, phase coherence, and elastic scattering, respec-tively. �σ (B) = [σ (B) + σ (−B)]/2−σ (B = 0 T) representsthe normalized conductivity in the unit of e2/πh.σ (B) can be calculated by the Drude model through thefollowing relations:σxx(B) = ρxx(B)/(ρxx2(B) + ρxy2(B)),where ρxx and ρxy represent the longitudinal and transverseresistivities, respectively. In 2D systems, ρxx(xy) is given byρxx(xy) = Rxx(xy)WL,where Rxx and Rxy represent the longitudinal and transverseresistances, respectively. W and L are the width (8.5 µm) andlength (3.0 µm) of the characterized channel in MoS2.BSO, Bφ, and Be can be determined by the spin-relaxationlength (LSO), phase coherence length (Lφ), and mean free path013216-6OBSERVATION OF WEAK LOCALIZATION IN … PHYSICAL REVIEW RESEARCH 6, 013216 (2024)(Le), respectively:BSO = h̄/(4eLSO2),Bφ = h̄/(4eLφ2),Be = h̄/(2eLe2).Lφ is extracted by the fitting the WL shown inthe main text, and Le can be calculated by Le =σ (B = 0 T)h̄/e2√4π/(gvns), where gv = 2 is the valley de-generacy in each spin-valley locked band and ns is totalcarrier density. At ns = 3.3×1012 cm−2, we obtain Lφ = 60nm (Bφ = 0.05 T) and Le = 14 nm (Be = 0.84 T).We then simulate how the SOC plays a role on the turnoverfrom WL to WAL. 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